
theorem
  for S, T being up-complete lower-bounded non empty Poset st [:S,T:]
  is continuous holds S is continuous & T is continuous
proof
  let S, T be up-complete lower-bounded non empty Poset such that
A1: for x being Element of [:S,T:] holds waybelow x is non empty directed and
A2: [:S,T:] is up-complete satisfying_axiom_of_approximation;
  hereby
    hereby
      set t = the Element of T;
      let s be Element of S;
A3:   waybelow [s,t] is directed by A1;
      [:waybelow s,waybelow t:] = waybelow [s,t] & proj1 [:waybelow s,
      waybelow t:] = waybelow s by Th44,FUNCT_5:9;
      hence waybelow s is non empty directed by A3,YELLOW_3:22;
    end;
    thus S is up-complete;
    thus S is satisfying_axiom_of_approximation
    proof
      set t = the Element of T;
      let s be Element of S;
      waybelow [s,t] is directed by A1;
      then ex_sup_of waybelow [s,t], [:S,T:] by WAYBEL_0:75;
      then
A4:   sup waybelow [s,t] = [sup proj1 waybelow [s,t],sup proj2 waybelow [s
      ,t]] by Th5;
      thus s = [s,t]`1
        .= (sup waybelow [s,t])`1 by A2
        .= sup proj1 waybelow [s,t] by A4
        .= sup waybelow [s,t]`1 by Th46
        .= sup waybelow s;
    end;
  end;
  hereby
    set s = the Element of S;
    let t be Element of T;
A5: waybelow [s,t] is directed by A1;
    [:waybelow s,waybelow t:] = waybelow [s,t] & proj2 [:waybelow s,
    waybelow t:] = waybelow t by Th44,FUNCT_5:9;
    hence waybelow t is non empty directed by A5,YELLOW_3:22;
  end;
  set s = the Element of S;
  thus T is up-complete;
  let t be Element of T;
  now
    let x be Element of [:S,T:];
    waybelow x is non empty directed by A1;
    hence ex_sup_of waybelow x,[:S,T:] by WAYBEL_0:75;
  end;
  then
A6: sup waybelow [s,t] = [sup proj1 waybelow [s,t],sup proj2 waybelow [s,t
  ]] by Th5;
  thus t = [s,t]`2
    .= (sup waybelow [s,t])`2 by A2
    .= sup proj2 waybelow [s,t] by A6
    .= sup waybelow [s,t]`2 by Th47
    .= sup waybelow t;
end;
